The correlation of the returns of the market (specifically the S&P 500 Index) and the VIX is large and negative, sampling in the region −85% to −100%. Without the results of the prior post, we are forced to seek a behavioural root for this observation:

demand for hedging instruments increases when the market is falling, increasing implied volatilities as the price of put options increases beyond fair value.

i.e. We have ascribed the observed covariance as a consequence of market participants' irrationality, as they believe that it's worth hedging only after they've lost money. However, Completely Asymmetric GARCH, such as that exhibited in the prior post, describes a process in which increases in future actual volatilities are only due to downwards moves and volatilities decrease after upwards moves. This linkage describes negative correlation between the returns of the index and future volatility and also describes a process which cannot crash upwards, as many commentators observe that the market does not do.

Why is this important? Well, consider if one believes that the increase in options prices on a down day is due to market participants situational bias. i.e. Their irrational belief that because the market is falling today then it is more likely to fall tomorrow. Under this scenario, the observed phenomenology represents an alpha or a predictable conditional mean for the future distribution of option trading profits. However, if the observations are consistent with the actual empirical price process — that down days do tend to increase future volatility — then there is no alpha. The increased prices represent an increase in fair value.

Under the first scenario, the right thing to do is to sell options; whereas, under the second, the right thing to do is to buy options. 

Market commentators frequently use the word “volatility” to mean that the market is moving downwards. To a financial economist, this is an error. Volatility means the stochastic element to price changes and it is not a synonym for the proportion of the distribution below the mean. In his Nobel Prize lecture, Harry Markowitz suggested the use of what he termed semi-variance to assess risk, a definition of risk that concentrates solely on the downsize and which is more in line with the usage of the commentator than the statistician.

In our prior two posts, discussing Asymmetric GARCH and the VOLVIX process, we found evidence of an option-like non-linear response of the variance to quadratic stimuli. In this post we look for a similar structure in a less esoteric series — the daily returns of the S&P 500 Index.

Applying a AGARCH(1,1) GJR model to the daily returns of the S&P 500 Index, but assuming a fundamentally leptokurtotic driving process, in this instance draws from Student's t Distribution, we find a model that provides an acceptibly good description of the data (the results of which are presented below).

Fit of AGARCH(1,1) Model to S&P 500 Index - Estimation by BFGS
Convergence in    31 Iterations. Final criterion was  0.0000028 <=  0.0000100
Daily(5) Data From 2000:01:03 To 2010:05:24
Usable Observations   2333
Log Likelihood                    7299.94328188

   Variable           Coeff       Std Error      T-Stat     Signif
*********************************************************************
1.  Mean             0.00008427   0.00017596      0.47891  0.63200177 ← mean daily return
2.  C                0.00000095   0.00000029      3.27083  0.00107232 ← static variance term
3.  A               -0.01428908   0.00888489     -1.60825  0.10778138 ← ARCH MA(1) parameter
4.  B                0.93586512   0.01186219     78.89480  0.00000000 ← GARCH AR(1) parameter
5.  D                0.14418349   0.01892191      7.61992  0.00000000 ← asymmetry parameter
6.  Shape           10.98934650   2.22903583      4.93009  0.00000082 ← distributional kurtosis 

This model strongly likes the downside response term in the variance and almost completely rejects its symmetric counterpart. The MLR test statistic for the inclusion of the asymmetry term in the model is χ²(1) = 78.2, which is very significant.

So, in conclusion, we can state that yes, as far as the variance response is concerned, the usage of the term “volatility” as a synonym for “downside risk” seems to be confirmed.

To complement what we've just learned about the asymmetry in the VOLVIX process, here's a quick chart illustrating the peculiar option-like response of VOLVIX to changes in VIX itself.

This chart was computed from data as of the close, 05/17/2010.

In the prior posts we have noted that the VIX index itself is extraordinarily volatile. Going back to the series for the start of this month, we see that the VIX increases from around 20 points on 05/03/2010 to over 40 points on 05/07/2010 — doubling in the space of a week — and then drops by more than 30% to then end of 05/10/2010. Even fundamentally leptokurtotic GARCH has difficulties explaining these moves, as exhibited by the χ²⁄d.o.f. in our histogram of the homogenized innovations of the process taking the value of 193⁄97.

One model that is often discussed for financial time series is the Asymmetric GARCH process, such as that defined by:

Here I is an indicator function that is zero if its argument is non-negative and unity otherwise. This is the model of Glosten, et al., known as GJR. This specifies as piecewise quadratic response of the variance to the innovation, conditioned on it's sign. This particular specification is non-degenerate with respect to B when D is fixed, which allows us to use the maximum likelihood ratio test to examine the significance of a non-zero parameter.

Performing this regression, we find a m.l.r. statistic of χ²(1) = 11.670634 with significance level 0.00063494, indicating strong desired of the data to include this term. The analysis chart does not look much different from the prior one, so I won't include a redundant copy, but the χ²⁄d.o.f. for the histogram of homogenized innovations decreases to 165⁄97. The regression results are:

GARCH Model - Estimation by BFGS
Convergence in    26 Iterations. Final criterion was  0.0000000 <=  0.0000100
Daily(5) Data From 2001:01:02 To 2010:05:17
Usable Observations   2172
Log Likelihood                    3205.25327235

   Variable                     Coeff       Std Error      T-Stat     Signif
*******************************************************************************
1.  Mean                     -0.002781284  0.001104296     -2.51860  0.01178209
2.  C                         0.000208902  0.000046653      4.47779  0.00000754
3.  A                         0.131878668  0.022583630      5.83957  0.00000001
4.  B                         0.860413599  0.021960624     39.17983  0.00000000
5.  D                        -0.122355877  0.034771870     -3.51882  0.00043347
6.  Shape                     5.361237381  0.572901025      9.35805  0.00000000

Here the estimated D almost exactly cancels the A, indicating that the volatility of the VIX increases when the value of the index increases, but that it decreases when the index decreases. This is an extreme asymmetry — a severe negative shock does not increase VOLVIX at all and provides the conditions for the runaway trend followed by an abrupt reversal we found in the recent data.

For a while now I've been publishing daily volatility estimates for the VIX index as a page on this blog. This is built from a simple GARCH(1,1) model based on the relative changes. i.e.

In addition to the conditional variance structure above, we use a fundamentally leptokurtotic, but symmetric, innovation so that

where GED represents the Generalized Error Distribution and Student represents Student's t Distribution. In both cases the ν parameter controls the kurtosis of the distribution. This model tracks the conditional variance of the VIX index reasonably well and, for the cast of the Student's t Distribution is exhibited in the model summary chart below.

Now one of the recurring themes on this blog is that introducing simple GARCH models and fundamentally leptokurtotic innovations goes a very long way towards restoring the analytical tractability of financial data. However, the innovations exhibited above appear to the eye to be much less well described by the given p.d.f. than for regular series, such as the S&P 500. One only has to examine the series of annualized volatilities to find some extraordinary values. At the time of writing the estimate computed on Monday was over 270% on an annualized basis. (A snippet of this data is tabulated below.) This would imply a catastrophic drawdown of the VIX to zero should be pretty likely — yet this is clearly forbidden by the true dynamics of the series. There must be more complicated behaviour embedded in this series. We shall examine some of these extensions in future posts.

The early estimate of the return of the Dynamic Trading Risk Factor is now available; the data can be downloaded from the blog.